Performance Modeling Single Queues CS 700. Acknowledgement

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1 Performane Modeling ingle Queue C 700 Aknowledgement Thee lide are baed on reentation reated and oyrighted by Prof. Daniel Menae GMU

2 Puroe of Model Provide a way to derive erformane metri from model arameter. Examle of erformane metri: Reone time Throughut Availability Tye of arameter: Workload intenity e.g., arrival rate ervie demand. 3 Tye of Model imulation: mimi flow of tranation through a ytem. Ditribution-driven Trae-driven Analyti: et of formula or omutational algorithm. Exat Aroximate Hybrid 4

3 When to Ue? Ue Exat Analyti Model Whenever Poible. Ue Aroximate Analyti Model: For firt-ut analyi If validated by imulation To redue ombination of inut arameter to imulation model. Ue imulation: If there i no tratable analyti model. 5 ingle Queue W T utomer waiting line T W + 6 3

4 Bakground: tohati Proee A tohati roe i a family of random variable {Xt t T}, defined on a given robability ae, indexed by the arameter t, where t varie over the index et T The value aumed by the random variable Xt are alled tate If tate ae i direte, then the tohati roe i a direte-tate roe, often referred to a a hain, otherwie it i a ontinuou-tate roe If the index et i direte, the roe i alled a direte arameter roe, otherwie it i a ontinou arameter roe 7 tohati roee ont d Conider a ingle-erver queue. We an identify everal tohati roee N k - number of utomer in the ytem at the time of dearture of the kth utomer. {N k k,, } i a direte arameter, direte-tate roe Xt - number of utomer in the ytem at time t {Xt 0 < t < } i a ontinuou arameter, direte tate roe W k - time the kth utomer ha to wait to reeive ervie {W k k,, } i a direte arameter, ontinuou tate roe Yt - umulative ervie requirement of all job in the ytem at time t {Yt 0 < t < } i a ontinuou arameter, ontinuou tate roe 8 4

5 tohati roee - ome tye Markov roe/hain -- if the future tate of a roe are indeendent of the at and deend only on the urrent tate, the roe i alled a Markov roe Birth-death roee -- direte tate Markov roee in whih tranition are retrited to neighboring tate only Poion roe -- if the interarrival time at a queue are IID indeendent and identially ditributed and exonentially ditributed, the arrival roe i alled a Poion roe Thi i beaue the number of arrival over a given interval of time will have a Poion ditribution 9 Examle of An Analyti Model: M/G/ Queue ingle erver. Arrival roe i Poion interarrival time are exonentially ditributed. ervie time i arbitrarily ditributed. Where λ E[ E[ + C T E[ + E[ + λe[ < 0 5

6 Little Law N X R The average number of utomer in a blak box i equal to the average time ent in the box multilied by the throughut of the box. N R X Little Law Examle I An NF erver wa monitored during 30 min and the number of I/O oeration erformed during thi eriod wa found to be 3,400. The average number of ative requet N req wa 9. What wa the average reone time er NF requet at the erver? 6

7 Little Law Examle I An NF erver wa monitored during 30 min and the number of I/O oeration erformed during thi eriod wa found to be 3,400. The average number of ative requet N req wa 9. What wa the average reone time er NF requet at the erver? blak box NF erver X erver 3,400 /,800 8 requet/e R req N req / X erver 9 / e 3 Little Law Examle II A large ortal ervie offer free ervie. The number of regitered uer i two million and 30% of them end end mail through the ortal during the eak hour. Eah mail take 5.0 e on average to be roeed and delivered to the detination mailbox. During the buy eriod, eah uer end 3.5 mail meage on average. The log file indiate that the average ize of an meage i 7,0 byte. What hould be the aaity of the ool for outgoing mail during the eak eriod? 4 7

8 Little Law Examle II A large ortal ervie offer free ervie. The number of regitered uer i two million and 30% of them end end mail through the ortal during the eak hour. Eah mail take 5.0 e on average to be roeed and delivered to the detination mailbox. During the buy eriod, eah uer end 3.5 mail meage on average. The log file indiate that the average ize of an meage i 7,0 byte. What hould be the aaity of the ool for outgoing mail during the eak eriod? AvgNumberOfMail Throughut x ReoneTime,000,000 x 0.30 x 3.5 x 5.0 / 3,600,96.7 mail AvgoolFile,96.7 x 7,0 byte 9.8 MByte 5 Little Law Examle III A Web-baed brokerage omany run a three-tiered ite. The ite i ued by. million utomer. During the eak hour, 0,000 uer are logged in imultaneouly. The e- ommere ite roee 3.6 million buine funtion er hour on a eak-load hour. What i the average reone time of an e-ommere funtion during the eak hour? 6 8

9 Little Law Examle III A Web-baed brokerage omany run a three-tiered ite. The ite i ued by. million utomer. During the eak hour, 0,000 uer are logged in imultaneouly. The e-ommere ite roee 3.6 million buine funtion er hour on a eak-load hour. What i the average reone time of an e-ommere funtion during the eak hour? Blak box E-ommere ite AverageReoneTime AvgNumberOfUer / itethroughut 0,000 / 3,600,000 / 3,600 0 e 7 Uing Little Law in the M/G/ Queue W E[ N q + C λ λ λ LINE Reoure T E[ N + C + λ λ λ LINE Reoure 8 9

10 Exerie Plot the reone time for M/G/ a a funtion of for M/M/, M/D/, and ditribution with oeffiient of variation equal to _ and. Aume that E[ 0.. Vary λ aordingly. What onluion do you take from looking at the grah? Average Reone Time Utilization M/D/ - C 0 M/G/ - C0.5 M/M/ - C M/G/ - C 0 0

11 M/G/, M/M/, and M/D/ M/G/: W E[ + C M/D/: W E[ M/M/: W E[ G/G/ Queue λe[ < 0

12 An Aroximation for G/G/ C a a C C + W + C + C / E[ : oeffiient of variation of the interarrival time. Aroximation i exat for M/G/, good for G/M/, and fair for G/G/. The aroximation imrove a inreae. 3 G/G/ Queue λe[ < 4

13 3 5 An Aroximation for G/G/ [ /, C a C E C W + Aroximation i exat for M/M/. The error inreae with C a and C. + 0!!! /, n n n C where i Erlang C formula. 6 The M/M/ Queue [ /, E C W + 0!!! /, n n n C where i Erlang C formula.

14 Additional Toi 7 M/G/ Buy Period Unfinihed work: Ut Ut inreae every time an arrival our by the ervie time of the arrival. Ut dereae at a rate of -. x x x3 x4 buy eriod idle eriod x5 x6 buy eriod t 8 4

15 M/G/ Buy Period Time elaed ine the erver beome buy until it beome idle again. E[ E[ B Exeted number of utomer erved in a buy eriod. E[ Nb 9 M/G/ With Vaation

16 M/G/ With Vaation The erver goe on vaation for a time V V i a generally ditributed r.v. one the erver beome idle. Aliation: olling ytem. E[ + C E[ V W + E[ V regular M/G/ waiting time 3 Examle of M/G/ with Vaation A ytem erve requet that arrive aording to a Poion roe at a rate of 0. requet/e. The requet roeing time harateriti are: E[ 3.5 e and C 0.3. When there no requet to be roeed, the ytem goe into elf-diagnoi mode, whih lat for an average of e with a oeffiient of variation of. After elf-diagnoi, the ytem goe bak to erve requet. If no requet are queued, a new elfdiagnoi mode i tarted. What i the average waiting time of a requet? 3 6

17 olution to M/G/ with Vaation Examle E[V W - M/G/ no Vaation W - M/G/ Vaation σ V E[ V σv E[ V E[ V E[ V E[ V E[ V + CV E[ V E[ V E[ V 33 M/G/ with Prioritie P tati rioritie,, P. P i the highet riority. FCF within eah riority queue. λ λ λ P 34 7

18 M/G/ with Non-Preemtive Prioritie W W Π 0 Π i P P P i λ E[ W0 Π λ E[ i P + 35 Examle of M/G/ with tati Prioritie A router reeive requet at a rate of l requet/e from a Poion roe. 50% of them are of riority, 30% of riority, and 0% of riority 3. Priority E[ e E[

19 4 Average Waiting Time Total Utilization W W W3 37 M/G/ with Preemtive Reume Prioritie T Π E[ P i Π i Π + P i Π λ E[ i + i / 38 9

20 Comaring Preemtive v. Non- Preemtive M/G/ Queue a b d e f gb*f Nonreemtive Preemtive Priority Lambda E[ Π E[ T T M/G/ Ditribution Reone Time: Lalae Tranform: L T L λ + λl ~ L E[ e ~ T LT E[ e x e f x dx x 0 t e ft t dt t

21 4 M/M/ Reone Time Ditribution E E L L L T λ λ λ + + [ / [ / [ / [ / E E L + Lalae Tranform of the reone time for an exonentially ditributed ervie time: 4 M/M/ Reone Time Ditribution E E L L L T λ λ λ + + [ / [ / [ / [ E t T e E t f Probability denity funtion for the reone time:

22 .d.f. for Reone Time for M/M/ ftt t 43 M/G/ Ditribution Waiting Time: Lalae Tranform: L W λ + λl ~ L E[ e ~ T LT E[ e x e f x dx x 0 t e ft t dt t 0 44

Lecture 2. Main Topics: (Part II) Chapter 2 (2-7), Chapter 3. Bayes Theorem: Let A, B be two events, then. The probabilities P ( B), probability of B.

Lecture 2. Main Topics: (Part II) Chapter 2 (2-7), Chapter 3. Bayes Theorem: Let A, B be two events, then. The probabilities P ( B), probability of B. STT315, Section 701, Summer 006 Lecture (Part II) Main Toics: Chater (-7), Chater 3. Bayes Theorem: Let A, B be two events, then B A) = A B) B) A B) B) + A B) B) The robabilities P ( B), B) are called